Short time existence on nonlinear parabolic PDE

Question

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of page and number of theorem ?

Answer 1

This is kind of meta-theorem. It has several version, and if someone was willing to write one theorem containing all the situations, it would be unreadible.

The fully non-linear case (example $\partial_t u=\det(\nabla^2u)$ with $u(t=0,\cdot)$ convex), is really involved. In the quasilinear case (example $\partial_t u={\rm div}(|\nabla u|^{p-1}\nabla u)$, it is still complicated, but several books treat it extensively.

The semi-linear case $\partial_tu+Lu=f(u,\nabla u)$, where $L$ is an elliptic linear operator (like the Laplacian $-\Delta$), has a rather simple philosophy. One follows the guidelines of the Cauchy-Lipschitz theory for ODEs. If $a\in X$ is the initial data, one rewrites the Cauchy problem as an integral equation $$u(t)=e^{-tL}a+\int_0^t e^{(s-t)L}f(u(s),\nabla u(s))ds=:Nu(t).$$ When $T>0$ is small enough and the Banach space $X$ is appropriate, one proves that $N$ is a contraction in some ball $B(a;r)$ of $C(0,T;X)$. Then Picard's theorem tells that there is a unique fixed point $u$ ; this is the local solution.

By appropriate, I mean that the operators $S_t:=e^{-tL}$ form a strongly continuous semi-group over $X$. In particular $S_t\in{\mathcal L}(X)$ and $$\|S_t\|_{{\mathcal L}(X)}\le Ce^{\omega t},\qquad\forall t>0$$ for suitable constants $C$ and $\omega$. In practice, we may deal with a scale of Banach spaces $X_s$, like the Sobolev spaces $H^s$, and we have $$\|S_t\|_{{\mathcal L}(X_s,X_r)}\le Ct^{\alpha(s-r)}e^{\omega t},\qquad\forall t>0$$ for some $\alpha>0$, which is related to the order of $L$.